Integrand size = 21, antiderivative size = 392 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a d e+\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {751, 837, 858, 733, 435, 430} \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (x \left (3 a e^2+4 c d^2\right )+a d e\right )}{6 a^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {c} \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {2 d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {c} \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}} \]
[In]
[Out]
Rule 430
Rule 435
Rule 733
Rule 751
Rule 837
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 d-\frac {3 e x}{2}}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx}{3 a} \\ & = \frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a d e+\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {\frac {1}{4} a c d e^2-\frac {1}{4} c e \left (4 c d^2+3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )} \\ & = \frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a d e+\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {d \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2}-\frac {\left (4 c d^2+3 a e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{12 a^2 \left (c d^2+a e^2\right )} \\ & = \frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a d e+\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {\left (\left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{6 \sqrt {-a} a \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 d \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} a \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {x \sqrt {d+e x}}{3 a \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a d e+\left (4 c d^2+3 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\left (4 c d^2+3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 12.12 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (4 c^2 d^2 x^3+a^2 e (d+5 e x)+a c x \left (6 d^2+d e x+3 e^2 x^2\right )\right )}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {(d+e x) \left (-\frac {2 e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c d^2+3 a e^2\right ) \left (a+c x^2\right )}{(d+e x)^2}+\frac {2 i \sqrt {c} \left (4 c^{3/2} d^3+4 i \sqrt {a} c d^2 e+3 a \sqrt {c} d e^2+3 i a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {c} e \left (4 c d^2+i \sqrt {a} \sqrt {c} d e+3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}\right )}{a^2 c e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right )}\right )}{12 \sqrt {a+c x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(320)=640\).
Time = 2.09 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.98
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 a \,c^{2} \left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (-\frac {\left (3 e^{2} a +4 c \,d^{2}\right ) x}{12 a^{2} \left (e^{2} a +c \,d^{2}\right ) c}-\frac {e d}{12 a \left (e^{2} a +c \,d^{2}\right ) c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {2 d}{3 a^{2}}-\frac {e^{2} d}{12 a \left (e^{2} a +c \,d^{2}\right )}-\frac {d \left (3 e^{2} a +4 c \,d^{2}\right )}{6 a^{2} \left (e^{2} a +c \,d^{2}\right )}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}-\frac {e \left (3 e^{2} a +4 c \,d^{2}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{6 \left (e^{2} a +c \,d^{2}\right ) a^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(776\) |
default | \(\text {Expression too large to display}\) | \(2292\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (2 \, a^{2} c d^{3} + 3 \, a^{3} d e^{2} + {\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{4} + 2 \, {\left (2 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (4 \, a^{2} c d^{2} e + 3 \, a^{3} e^{3} + {\left (4 \, c^{3} d^{2} e + 3 \, a c^{2} e^{3}\right )} x^{4} + 2 \, {\left (4 \, a c^{2} d^{2} e + 3 \, a^{2} c e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (a c^{2} d e^{2} x^{2} + a^{2} c d e^{2} + {\left (4 \, c^{3} d^{2} e + 3 \, a c^{2} e^{3}\right )} x^{3} + {\left (6 \, a c^{2} d^{2} e + 5 \, a^{2} c e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{4} c^{2} d^{2} e + a^{5} c e^{3} + {\left (a^{2} c^{4} d^{2} e + a^{3} c^{3} e^{3}\right )} x^{4} + 2 \, {\left (a^{3} c^{3} d^{2} e + a^{4} c^{2} e^{3}\right )} x^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d + e x}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
[In]
[Out]